math.sqrt on complex, real part

Average Error: 38.7 → 1.3

Time: 9.3s

Precision: binary64

Cost: 26372

Math

\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{-146}:\\ \;\;\;\;0.5 \cdot \left|\frac{im}{\sqrt{\mathsf{hypot}\left(re, im\right) - re}} \cdot \sqrt{2}\right|\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -1.5e-146)
   (* 0.5 (fabs (* (/ im (sqrt (- (hypot re im) re))) (sqrt 2.0))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
}

↓

double code(double re, double im) {
	double tmp;
	if (re <= -1.5e-146) {
		tmp = 0.5 * fabs(((im / sqrt((hypot(re, im) - re))) * sqrt(2.0)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
}

↓

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.5e-146) {
		tmp = 0.5 * Math.abs(((im / Math.sqrt((Math.hypot(re, im) - re))) * Math.sqrt(2.0)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))

↓

def code(re, im):
	tmp = 0
	if re <= -1.5e-146:
		tmp = 0.5 * math.fabs(((im / math.sqrt((math.hypot(re, im) - re))) * math.sqrt(2.0)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	return tmp

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
end

↓

function code(re, im)
	tmp = 0.0
	if (re <= -1.5e-146)
		tmp = Float64(0.5 * abs(Float64(Float64(im / sqrt(Float64(hypot(re, im) - re))) * sqrt(2.0))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return tmp
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
end

↓

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.5e-146)
		tmp = 0.5 * abs(((im / sqrt((hypot(re, im) - re))) * sqrt(2.0)));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_] := If[LessEqual[re, -1.5e-146], N[(0.5 * N[Abs[N[(N[(im / N[Sqrt[N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Target

Original	38.7
Target	33.6
Herbie	1.3

\[\begin{array}{l} \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if re < -1.50000000000000009e-146

   1. Initial program 51.7

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]

   2. Simplified 33.6

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      Proof
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (hypot.f64 re im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))))))): 2 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 51.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{2} - re \cdot re}{\mathsf{hypot}\left(re, im\right) - re}}} \]

   4. Taylor expanded in re around 0 31.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\mathsf{hypot}\left(re, im\right) - re}} \]

   5. Simplified 31.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im}}{\mathsf{hypot}\left(re, im\right) - re}} \]
      Proof
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (/.f64 (*.f64 im im) (-.f64 (hypot.f64 re im) re))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 im 2)) (-.f64 (hypot.f64 re im) re))))): 2 points increase in error, 0 points decrease in error

   6. Applied egg-rr 0.5

      \[\leadsto 0.5 \cdot \color{blue}{\left|\frac{im}{\sqrt{\mathsf{hypot}\left(re, im\right) - re}} \cdot \sqrt{2}\right|} \]

   if -1.50000000000000009e-146 < re

   1. Initial program 31.1

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]

   2. Simplified 1.8

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      Proof
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (hypot.f64 re im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))))))): 2 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{-146}:\\ \;\;\;\;0.5 \cdot \left|\frac{im}{\sqrt{\mathsf{hypot}\left(re, im\right) - re}} \cdot \sqrt{2}\right|\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.3
Cost	13700

\[\begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{-146}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\mathsf{hypot}\left(re, im\right) - re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 2
Error	11.8
Cost	13508

\[\begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+190}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \frac{\sqrt{2}}{\sqrt{re \cdot -2}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 3
Error	11.8
Cost	13508

\[\begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{+190}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \frac{im}{\sqrt{re \cdot -2}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 4
Error	11.3
Cost	13444

\[\begin{array}{l} \mathbf{if}\;re \leq -2.45 \cdot 10^{+199}:\\ \;\;\;\;0.5 \cdot {\left(\frac{-1}{\frac{\frac{re}{im}}{im}}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 5
Error	26.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -3.5 \cdot 10^{-219}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 1.86 \cdot 10^{-65}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 6
Error	26.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -3.6 \cdot 10^{-207}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 6 \cdot 10^{-65}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 7
Error	27.2
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -4.1 \cdot 10^{-219}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 4 \cdot 10^{-64}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 8
Error	37.0
Cost	6852

\[\begin{array}{l} \mathbf{if}\;im \leq 1.15 \cdot 10^{-64}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 9
Error	47.4
Cost	6720

\[0.5 \cdot \sqrt{im \cdot 2} \]

Reproduce

herbie shell --seed 2022343 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))